System and Method for Disintegrated Channel Estimation in Wireless Networks

ABSTRACT

A system and method for disintegrated channel estimation in wireless networks. The system provides a disintegrated channel estimation technique required to accomplish the spatial diversity supported by cooperative relays. The system includes a filter-and-forward (FF) relaying method with superimposed training sequences for separately estimating the backhaul and the access channels. To reduce inter-relay interference, a generalized filtering technique is provided which multiplexes the superimposed training sequences from different relays to the destination by time-division multiplexing (TDM), frequency-division multiplexing (FDM) and code-division multiplexing (CDM) methods.

RELATED APPLICATIONS

This application claims priority to U.S. Provisional Patent ApplicationNo. 62/277,704 filed on Jan. 12, 2016, the entire disclosure of theapplication hereby expressly incorporated by reference.

STATEMENT OF GOVERNMENT INTERESTS

This invention was made with government support under Grant No.CNS-1456793 and Grant No. ECCS-1343210 awarded by the National ScienceFoundation. The government has certain rights in the invention.

BACKGROUND OF THE INVENTION

Field of the invention

The present disclosure relates to channel estimation in wirelessnetworks. More specifically, the present disclosure relates to a systemand method for disintegrated channel estimation in wireless networks.

Related Art

In the field of wireless telecommunications, providing communicationschannels that are reliable and of high quality of service are paramountconcerns. Cooperative communication has become one of the most essentialaspects of recent mobile communication. A relay network represents oneform of cooperative communication and has attracted a large amount ofresearch in recent years. A space-time coding technique supported byrelays has, in the past, been employed to achieve spatial diversity. Theachievable diversity obtained using cooperative networks was studiedpreviously. It has been found that the relay nodes alter the receivedsignals according to the space-time block code (STBC) arrangement andthen send the resultant signals to the destination node. The diversityachieved using the distributed STBC (D-STBC) with relays can be achievedby accurately estimating multiple timing errors and multiple frequencyoffsets as well as acquiring precise knowledge of the channel stateinformation (CSI). Additionally, previous studies have successfullyaddressed the problems caused by timing errors and frequency offsets.

To acquire accurate knowledge of CSI, several channel estimationtechniques have been investigated previously. These techniques are oftencategorized as (1) cascaded channel estimation; and (2) disintegratedchannel estimation. However, cascaded channel estimation is ineffectivefor several applications such as optimal relay matrix design andbeamforming. In existing relay beamforming techniques, relay gains wereadjusted to suppress interference. Concerning disintegrated channelestimation, two methods are usually adopted to obtain disintegrated CSIat the destination node. In the first method, the source-relay channelsare estimated at the relay nodes first by means of a conventionalsingle-hop channel estimation, and the estimates are then quantized,compressed and sent toward the destination node via dedicated channels.Thus, the relay nodes are required to conduct source-relay channelestimation and compression with high complexity, and additional timeslots or subchannels are required to deliver the source-relay estimates,decreasing both the spectral and energy efficiencies. In the secondmethod, both the source-relay and the relay-destination channels areestimated at the destination node. A method utilizing relay pilots toconduct disintegrated channel estimation was studied previously, inwhich several pilot subcarriers are preserved for sending the relaypilots. It was found that the destination node can estimate thesource-relay-destination and the relay-destination channels byexploiting the source and the relay pilots, respectively. Theorthogonality requirement of the relay pilots was addressed in aprevious study. In addition, the orthogonality was exploited to generatetraining sequences that can be applied in an amplify-and-forward (AF)relay network. Because prior efforts only studied a disintegratedchannel estimation in a single-relay network, generalizations andimprovements of the disintegrated channel estimation techniques inmultiple cooperative relay network are strongly desired.

Accordingly, what would be desirable is a system and method fordisintegrated channel estimation in wireless networks, which addressesthe foregoing shortcoming of existing systems.

SUMMARY OF THE INVENTION

The present disclosure relates to a system and method for disintegratedchannel estimation in wireless networks. The system provides adisintegrated channel estimation technique required to accomplish thespatial diversity supported by cooperative relays. The system includes afilter-and-forward (FF) relaying method with superimposed trainingsequences for separately estimating the backhaul and the accesschannels. To reduce inter-relay interference, a generalized filteringtechnique is provided. Unlike the interference suppression methodcommonly used in conventional FF relay networks, the generalizedfiltering matrix provided herein multiplexes the superimposed trainingsequences from different relays to the destination by time-divisionmultiplexing (TDM), frequency-division multiplexing (FDM) andcode-division multiplexing (CDM) methods. The Bayesian Cramér-Rao lowerbounds (BCRBs) are derived as the estimation performance benchmark. Themean square errors (MSEs) of the disintegrated channel estimation arealso derived. Finally, the improvements offered by the present systemare verified by comprehensive computer simulations in conjunction withcalculations of the BCRBs and the MSEs.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing features of the invention will be apparent from thefollowing Detailed Description, taken in connection with theaccompanying drawings, in which:

FIG. 1 is a flowchart illustrating processing steps carried out by thesystem of the present disclosure;

FIG. 2 is a diagram illustrating a relay network in which the system ofthe present disclosure can be implemented;

FIG. 3 is diagram illustrating a pilot structure implemented in awrieless network having 2 relays are arranged in parallel;

FIG. 4 is a diagram illustrating a subcarrier arrangement of anFF/FDM-SITS training slot in a 2-parallel-relay network;

FIG. 5 is a diagram illustrating the slot structure of the CDM approachof the present disclosure in a 2-parallel-relay network;

FIGS. 6-12 are diagrams illustrating simulation results performed inconnection with the system of the present disclosure; and

FIG. 13 is a diagram illustrating hardware components capable ofimplementing the system of the present disclosure.

DETAILED DESCRIPTION OF THE INVENTION

The present disclosure relates to a system and method for disintegratedchannel estimation in wireless networks, as disclosed herein in detailin connection with FIGS. 1-13.

A generic relaying signal model is disclosed herein to effectivelyrepresent the AF and filter-and-forward (FF) relaying variants. Thesystem can oeprate with multiple relays arranged in a parallel manner toaccomplish D-STBC. Orthogonal training sequences are generated to assistthe destination node in estimating all disintegrated channels. Multi-hoprelaying can also be performed by utilizing the superimposed trainingmethods. Rather than choosing the relay gain to minimize the inter-linkinterference between the source-relay and the source-destination links,the sytem of the present disclosure applies a generalized filteringmatrix as a kernel to multiplex the relay pilot sequences prior to therelay-destination transmission. The generalized filtering method of thepresent disclosure can coordinate the relay pilot sequences byexploiting the time-division multiplexing (TDM), the frequency-divisionmultiplexing (FDM), or the code-division multiplexing (CDM) methods. Thesystem can effectively overcome inter-relay interference (IRI), therebyreducing both the mean square errors (MSEs) of the disintegrated channelestimation and the symbol-error rate (SER) obtained using the STBC inthe relay network. The channel estimation technique of the presentdisclosure can be effectively applied in a scalable FF relay network,especially when a relay selection mechanism is applied or a relayoccasionally joins/leaves the currently operating network withoutchanging the bi-phase protocol.

The protocol of the present disclosure can be briefly described asfollows. The mobile unit can detect several relays as candidates tosupport the cooperative communication and then select the requiredrelays to establish STBC. When a relay occasionally joins (or leaves)the currently operating network, the mobile unit can stack it in (or popone up from) the candidate list without informing the base stationbecause the protocol and the frame structure are unchanged.

Some mathematical notations frequently used herein are summarized asfollows. Matrices (uppercase) and vectors (lowercase) are in boldface.C^(n×m) in and C^(m) represent the complex matrices of size n×m andcomplex vectors of size m, respectively. The superscripts (•)*, (•)^(T),(•)^(H) and (•)⁻¹ represent the complex conjugate, the vector (ormatrix) transpose, the Hermitian and the inverse operator, respectively.Φ(•) denotes a circulant function that converts a vector in C^(n) into acirculant matrix in C^(n×n); it aids in representing the circularconvolution of the signal and the channel impulse response (CIR)vectors. Θ(•) denotes a diagonal matrix generating function thatallocates the parameter vector into the diagonal terms of a squarematrix, Θ⁻¹(•) denotes an inverse matrix of the diagonal matrix. F_(N)represents the N-point discrete Fourier transform (DFT) matrix with (p,q)-entry expressed as

${{\langle F_{N}\rangle}_{p,q} = \frac{\omega^{pq}}{\sqrt{N}}},$

where p, q=0,1, . . . , N−1 and

$\omega = {e^{{- j}\frac{2\pi}{N}}.}$

Frequency domain (FD) signals are represented as ^(˜)(•), e.g., {tildeover (x)}=F_(N)x. Additionally, it is noted that the training slotlength employed herein is the duration of an OFDM symbol, not a slotthat contains 7 OFDM symbols, as defined in the LTE specification. Theterms “slot” and “subslot” are employed to avoid the confusion thatmight arise from the use of the terms “symbol” and “subsymbol.”

Reference is now made to FIGS. 1-2 of the present disclosure, whereinFIG. 1 is a flowchart represnting overall processing steps carried outby the system of the present disclosure and FIG. 2 is a diagramillustrating a sample relay network in which the system of the presentdisclosure could be implemented. As depicted in FIG. 2, a typicaldual-hop multi-relay network consists of one source node denoted as

, M relay nodes, and one destination node denoted as

. The kth relay node is denoted as

_(k), where k=1,2, . . . , M. The direct link is absent or is omitted inFIG. 2 solely to focus on the multi-relay channel estimation because therelay-based spatial diversity is usually required when the

-

signal-to-noise ratio (SNR) is very low such that the

-

link can be neglected. A quasi-static multipath channel that can usuallybe represented as a tapped-delay-line (TDL) model is considered here.The channel tap-weighting coefficients are assumed to be constant withina time slot but to vary between slots. The zero-padded CIR vectors areused to give the transmitted signal and CIRs the same length N. The

-

_(k) CIR vector is denoted as h_(k)=[{hacek over (h)}_(k) ^(T)0_(1×(N−L)_(h) ₎]^(T) ∈ C^(N), where {hacek over (h)}_(k) ∈ C^(L) ^(h) is the trueCIR vector with length L_(h). The zero-padded

_(k)-

CIR vector is defined as g_(k)=[{hacek over (g)}_(k) ^(T)0_(1×(N−L) _(g)₎]^(T), where the true CIR vector {hacek over (g)}_(k) is of lengthL_(g). The zero-padded

-

_(k)-

CIR vector is represented as v_(k)=[{hacek over (v)}_(k) ^(T)0_(1×(N−L)_(h) _(−L) _(g) ₊₁₎]^(T) where the true CIR vector {hacek over (v)}_(k)is of the effective channel length L_(v)=L_(h)+L_(g)−1. Based on cyclicprefix (CP) orthogonal-frequency-division-multiplexing (OFDM)techniques, the channel v_(k) can be obtained through the circularconvolution of h_(k) and g_(k) with v_(k)=Φ(h_(k))g_(k)=Φ(g_(k))h_(k).Furthermore, the circulant matrix converted from v_(k) can be written asΦ(v_(k))=Φ(h_(k))Φ(g_(k)). Although the previous equation holds forL_(h)+L_(g)−1≦N, in practice, a stricter requirement of L_(h)+L_(g)−1exists: L_(h)+L_(g)−1 must be less than the CP length in the OFDMcommunication systems to avoid inter-symbol interference (ISI).

Referring to FIG. 1., a flowchart 10 illustrates the overall processsteps carried out by the system of the present disclosure. In describingthe steps of flowchart 10 of FIG. 1, reference is also made to FIG. 2.In step 12, the source node

of FIG. 2 transmits a pilot signal and a data signal to the relay nodes

₁ . . .

_(k) of FIG. 2. Then, in step 14, the relay nodes

₁ . . .

_(k) receive and filter the pilot signal. In step 16, the relay nodes

₁ . . .

_(k) process the pilot signal and generate an orthogonal pilot signal.Next, in step 18., the relay nodes

₁ . . .

_(k) superimpose the orthogonal pilot signal onto the filtered pilotsignal. In step 20, the relay nodes

₁ . . .

_(k) transmit the superimposed and filtered pilot signals to thedestination node

of FIG. 2. Then, in step 22, the destination node

receives the superimposed and filtered pilot signals, and estimates thedisintegrated channel information using the superimposed and filteredpilot signals. In step 24, the relay nodes

₁ . . .

_(k) of FIG. 2 each process the data signal a into space-time coded datasignal and transmit the space-time coded data signal to the destinationnode

. Finally, in step 26, the destination node

receives the spce-time coded data signals from the relay nodes

₁ . . .

_(k) and decodes the space-time coded data signals using the estimateddisintegrated channel information.

The processing steps discussed above in connection with FIGS. 1-2 willnow be described in greater detail. It is noted that, in the presentdisclosre, a wireless CP-OFDM communication system is considered havinga block-type pilot arrangement in which training signals are transmittedperiodically via all subcarriers.

The basic transmission protocol of a relay network is often composed oftwo phases. In the first phase, the relay nodes receive the signaltransmitted from the source node. The received signal on the kth relaycan be written as:

y _(k)=√{square root over (P _(s))}Φ(h _(k))u _(S) +n′ _(k),   (1)

where u_(s) is the training signal transmitted from

, P_(s) denotes the transmitted power of the training signal, Φ(h_(k))denotes the circulant CIR matrix of the

-

_(k) link, and n′_(k) is an additive white Gaussian noise (AWGN)introduced at

_(k). In the second phase, any individual relay

_(k) sends a signal to node

with or without further signal processing, depending on the relayingstrategy. The generic relaying model considered herein can berepresented in the form of a matrix linear equation, and the receivedsignal at

can be expressed as:

y _(D)=Σ_(k=1) ^(M)[Φ(g _(k))(W _(k)y_(k) +u _(k))]+n _(D),   (2)

where Φ(g_(k)) is the circulant CIR matrix of the

_(k)-

link and n_(D) denotes the AWGN at

. A general filtering matrix W_(k) and a superimposed signal vectoru_(k) are jointly employed in (2) to consider various relayingstrategies.

In the information-bearing transmission, D-STBC are employed to achievethe spatial diversity. To utilize the D-STBC technique in relaynetworks, the symbols on the relay nodes should be altered in accordancewith its cooperative pattern. For example, considering a 2 -relaynetwork, the relay should be capable of storing two symbols to conductthe rearrangement. In the first phase,

_(k) receives the signal from

on the mth symbol block,

y _(k,m)=Φ(h _(k))x _(m) +n _(k,m),   (3)

where k and m denote the relay index and the stored symbol index,respectively, and x_(m), is the mth data block transmitted from

. The coding matrix can be expressed as:

$\begin{matrix}{Y_{R} = {\begin{bmatrix}y_{1,1} & {- y_{2,2}^{*}} \\y_{1,2} & y_{2,1}^{*}\end{bmatrix}.}} & (4)\end{matrix}$

If a set of data signals transmitted from

is [x₁ ^(T)x₂ ^(T)]^(T), then by using equation (3) and the STBCarrangement in equation (4) and taking the complex conjugate on thesecond symbol of the received signal, the signal received in

can be expressed as:

$\begin{matrix}{\begin{bmatrix}y_{D,1} \\y_{D,2}^{*}\end{bmatrix} = {\begin{bmatrix}{{\alpha_{1}{\Phi ( g_{1} )}{\Phi ( h_{1} )}} - {\alpha_{2}{\Phi ( g_{2} )}{\Phi^{*}( h_{2} )}}} \\{\alpha_{2}{\Phi ( h_{2} )}{\Phi^{*}( g_{2} )}\alpha_{1}{\Phi^{*}( h_{1} )}{\Phi^{*}( g_{1} )}}\end{bmatrix}{{\quad\quad}\lbrack \begin{matrix}x_{1} \\x_{2}^{*}\end{matrix} \rbrack} {\quad{{+ \lbrack {{\quad\quad}\begin{matrix}{{\alpha_{1}{\Phi ( g_{1} )}n_{1,1}} - {\alpha_{2}{\Phi ( g_{2} )}n_{2,2}^{*}}} \\{{\alpha_{1}{\Phi^{*}( g_{1} )}n_{1,2}^{*}} + {\alpha_{2}{\Phi^{*}( g_{2} )}n_{2,1}}}\end{matrix}} \rbrack} + {\begin{bmatrix}n_{D,1} \\n_{D,2}^{*}\end{bmatrix}.}}}}} & (5)\end{matrix}$

Because the relay nodes herein are assumed to be unable to acquire

-

_(k) CSI through channel estimation, the filtering matrix W_(k) in thedata transmission duration is thus chosen as an identity matrix with aconstant gain α_(k)I, which can be considered as a fixed-gain AFrelaying scenario.

To detect the STBC signal, the matrix in the first term of equation (5)should be evaluated first. However, the off-diagonal terms containingΦ(g₂)Φ*(h₂) and Φ(h₂)Φ*(g₂) cannot be obtained directly from equation(2). Therefore, the STBC technique unavoidably requires disintegratedchannel estimation. To keep the relay complexity low, a relay does notestimate the

-

_(k) channel, and

is the only node to conduct channel estimation.

A technique that effectively maintains the currently existing protocols,spectral efficiency, and network scaleability in multi-relay networks ishighly desired, and its development is one goal of the presentdisclosure. From equation (2), if the filtering matrix W_(k) issubstituted by α_(k)I, the signal model of the AF relaying strategyassisted from the superimposed training sequences can be obtained as:

y _(D=)√{square root over (P _(s))}Σ_(k=1) ^(M)[α_(k)Φ(g _(k))Φ(h_(k))]u _(S)+Σ_(k=1) ^(M)√{square root over (P _(k))}Φ(g _(k))u_(k)+Σ_(k=1) ^(M)α_(k)Φ(g _(k))n _(k) +n _(D),   (6)

where α_(k) denotes the gain of

_(k). The individual

_(k)-

CIR vector g_(k) can be estimated at

via the corresponding relay pilot u_(k). The source pilot u_(s) can beemployed to estimate the compound

-

-

channel, v=Σ_(k=1) ^(M) [α_(k)Φ(g_(k))Φ(h_(k))], as in the first term ofequation (6). In a parallel multi-relay scenario, individual

-

_(k) CIR vectors, h_(k), k=1,2, . . . , M, are mixed with each other andcannot be separately estimated; thus, IRI is unavoidable. To generalizethe application of multi-relay networks, a novel filtering strategy toaddress IRI in conjunction with a superimposed training method isproposed.

The system of the present disclosure can be considered a FF relayingstrategy with superimposed training sequences (FF/SITS). By insertingthe received signal on the kth relay as written in equation (1) into(2), the received signal at

can be rewritten as y_(D)=√{square root over (P_(s))}Σ_(k=1) ^(M)Φ(g_(k))W_(k)Φ(h_(k))u_(S)+Σ_(κ=1) ^(M) √{square root over(P_(κ))}Φ(g_(κ)+Σ_(κ=1) ^(M) Φ(g_(κ))n_(κ)+n_(D), wheren_(κ)=W_(κ)n′_(κ). If W_(k) is chosen to be a circulant matrix withrelay gain α_(k), y_(D) can be reformulated as:

y _(D=)√{square root over (P _(s))}Σ_(k=1) ^(M) α_(k)Φ(g _(k))Φ(h _(k))u_(S,k)+Σ_(κ=1) ^(M) √{square root over (P _(κ))}Φ(g _(κ))u _(κ)+Σ_(K=1)^(M) α_(κ)Φ(g _(κ))n _(κ) +n _(D),   (7)

Hereafter, α_(k) is set to 1 in order to focus on the work of the filterW_(k), u_(S,k)=W_(k)u_(S) according to the relation:W_(k)Φ(h_(k))u_(S)=Φ(h_(k))W_(k)u_(S). In equation (7), the first termis the desired term for conducting the

-

-

channel estimation by means of u_(S,k); the second term is the desiredterm for conducting the

_(κ)-

estimation by means of u_(κ); the third term denotes the relay noisesexperiencing

_(κ)-

links; and the fourth term represents the destination noise. In thefiltering process, all the pilots must be chosen such that the followingorthogonality requirements are satisfied:

$\begin{matrix}{{{\frac{1}{N}A_{L}{\Phi^{H}( u_{k} )}u_{k}} = {e_{1}{\delta \lbrack {k - \kappa} \rbrack}}},{{\frac{1}{N}A_{L}{\Phi^{H}( u_{S,k} )}u_{S,k}} = {e_{1}{\delta \lbrack {k - \kappa} \rbrack}}},{{\frac{1}{N}A_{L}{\Phi^{H}( u_{S,k} )}u_{k}} = 0},} & (8)\end{matrix}$

where δ[•] represents the discrete-time Kronecker delta function, k, κ ∈{1,2, . . . , M} , A_(L)=[I_(L)0_(L×(N−L))]_(L×N) is a segmenting matrixthat consists of an identity matrix and a zero matrix, L denotes themaximum delay spread or may simply be set as the CP length to considerthe worst condition, and e₁=[10_(1×(N−1))]^(T) denotes the N×1elementary vector. In equation (8), the first orthogonal requirement isadopted to separate the channel estimations between g_(k) and g_(κ); thesecond orthogonal requirement is adopted to separate the channelestimations between v_(k) and v_(κ); and the third orthogonalrequirement is adopted to separate the channel estimations between thelinks of

-

_(k)-

and

_(κ)-

. The orthogonality requirements hold if the degrees of freedom of theCSI are assumed to be lower than the CP length. W_(k) is utilized toeffectively multiplex the CSIs on different links in either the timedomain (TD) or the FD to maintain the orthogonality. To obtain completeknowledge of the disintegrated channels, the maximum number of relaysunder the coordination of the proposed filtering method cannot exceedN/2L. The third orthogonality requirement is also met. Three primarymultiplexing methods, TDM, FDM and CDM, are applied here to the parallelmulti-relay scenario to satisfy the first and second orthogonalityrequirements.

The TDM approach equally partitions a single training time slot[1] intoM subslots, within any of which a relay node

_(κ) sends its own relay pilot u_(κ). To achieve the filter functiondescribed in equation (7), a filter arranges

-

_(k) channel information to a segmented orthogonal pilot ū_(S,k).Therefore, the kth relay filtering matrix can be formulated as:

W_(k)

D _(T,k) ^(H)Φ(ū_(S,k))A_(N) _(r) W_(FDLS),

k=1,2, . . . , M ,   (9)

where “

” represents “defined as”; u_(S,k)=F_(N) _(r) ^(H)D_(T,k)ũ_(S) asegmented version of the source pilot with length N_(r), N_(r) is thelength of u_(S,k); W_(FDLS)=F_(N) ^(H)Θ⁻¹(ũ_(S))F_(N); A_(N) _(r)=[I_(N) _(r) 0_(N) _(r) _(×(N−N) _(r) ₎]_(N) _(r) _(×N); and the matrixD_(T,k) consists of an N_(r)×N_(r) identity matrix and zeros in theremaining terms, i.e.:

D _(T,k) =[T ₁0_(N) _(r) _(×N) _(cp) T ₂ . . . 0_(N) _(r) _(×N) _(cp) T_(M)]_(N) _(r) _(×N)′  (10)

T_(j)=I_(N) _(r) δ[j−k], 0_(N) _(r) _(×N) _(cp) is employed to reserve aCP insertion, and zero padding may be needed if N>MN_(r)+(M−1)N_(cp). Inequation (9), W_(FDLS) conducts the FD least squares (LS) channelestimation of the

-

_(k) channel, and W_(FDLS) can be replaced by any

-

_(k) channel estimation, e.g., a TD LS channel estimation:W_(TDLS)=(Φ^(H)(u_(S))Φ(u_(S)))⁻¹Φ^(H)(u_(S)). The FD LS channelestimate is then segmented to N_(r) by A_(N) _(r) . For practicalconsiderations and to clarify the major contribution, the LS channelestimation is chosen. Using D_(T,k), the segmented channel estimate of

-

_(k) is then placed in the kth subslot. As an example, the pilotstructure employed in the scenario in which 2 relays are arranged inparallel is depicted in FIG. 3, which illustrates a training slotstructure of the FF/TDM-SITS in a 2-parallel-relay network. In FIG. 3,ũ_(j) denotes a segmented version of u_(j), i=S, 1,2; and ũ_(k) is asegmented relay pilot, which is orthogonal to ũ_(S,k) and to the otherrelay pilots. Each subslot preserves a CP interval to avoid theinter-block interference (IBI) caused by the multipath fading channel.Because the degree of freedom of the signal vector space is limited, thesegmenting length in TDM approach must meet the following requirements:N≧MN_(r)+(M−1)N_(cp), N_(r)≧2N_(cp). Because all matrices arewell-defined and pre-determined on

_(k), W_(k) in equation (9) can be pre-calcul ated and pre-stored.

At destination node

(see FIG. 2), the kth subslot of y_(D) is sifted by {tilde over(y)}_(D,k)=D_(T,k)y_(D), k=1,2, . . . , M. Using equation (8), g_(k) canbe estimated with:

$\begin{matrix}{{{\hat{g}}_{k} = {\frac{1}{N_{r}\sqrt{P_{k}}}A_{L,N_{r}}{\Phi^{H}( {\overset{\_}{u}}_{k} )}{\overset{\_}{y}}_{D,k}}},{k = 1},2,\ldots \mspace{14mu},M,} & (11)\end{matrix}$

where A_(L,N) _(r) =[I_(L)0_(L×(N) _(r) _(−L))]_(L×N) _(r) . Meanwhile,the channel estimation of the cascaded CIR vector v_(k) of

-

_(k)-

can be conducted by:

$\begin{matrix}{{{\hat{v}}_{k} = {\frac{1}{N_{r}\sqrt{P_{S}}}A_{L,N_{r}}{\Phi^{H}( {\overset{\_}{u}}_{S,k} )}{\overset{\_}{y}}_{D,k}}},{k = 1},2,\ldots \mspace{14mu},{M.}} & (12)\end{matrix}$

Using the above channel estimates, the

-

_(k) CIR vector h_(k) can be estimated using the LS criterion, i.e.:

ĥ _(k)=(Φ^(H)(ĝ _(k))Φ(ĝ _(k)))⁻¹Φ^(H)(ĝ _(k)){circumflex over (v)}_(k),

k=1,2, . . . , M.   (13)

As disclosed above, the channel estimates of different

-

_(k) links are allocated to the interleaved subslots for an TDMapproach. In effect, the FDM approach interleaves the channel estimatesof different

-

_(k) links onto different subcarriers. Therefore, the filtering matrixon the kth relay node can be formulated as:

W_(k)

F_(N) ^(H){tilde over (D)}_(F,k)F_(N), k=1,2, . . . , M ,   (14)

where {tilde over (D)}_(F,k) denotes the subcarrier sift matrix that isa diagonal matrix consisting 1's or 0's, i.e.,

$\begin{matrix}{{{\overset{\sim}{D}}_{F,k} = {\Theta \{ {d_{1}d_{2}\mspace{14mu} \ldots \mspace{14mu} d_{N}} \}}},{d_{i} = \{ {\begin{matrix}1 & {{{i{mod}}\; M} = k} \\0 & {{{i{mod}}\; M} \neq k}\end{matrix},} }} & (15)\end{matrix}$

mod denotes the modulo operator. The

-

_(k) CSI are projected to different dimensions by filtering the signalsent from

. A commonly used allocation method is to load individual relay pilotsonto equally spaced subcarriers. An example of the subcarrierarrangement in a 2-parallel-relay network is shown in FIG. 4, whichillustrates a subcarrier arrangement of an FF/FDM-SITS training slot ina 2-parallel-relay network. In FIG. 4, ũ_(S,k), k=1,2. . . , M denotesthe distributed source pilots, which are allocated to equally spacedsubcarriers, and ũ_(k), k=1,2, . . . , M denotes the kth FD relay pilot.

At destination node

(see FIG. 2), the individual

_(k) D CIR vector g_(k) can be estimated by:

$\begin{matrix}{{{\hat{g}}_{k} = {\frac{1}{N\sqrt{P_{k}}}A_{L}{\Phi^{H}( u_{k} )}y_{D}}},{k = 1},2,\ldots \mspace{14mu},{M.}} & (16)\end{matrix}$

The compound channel of the link

-

-

can be estimated as:

$\begin{matrix}{{\hat{v} = {\frac{1}{N\sqrt{P_{S}}}{\Theta ( A_{B} )}{\Phi^{H}( u_{S} )}y_{D}}},} & (17)\end{matrix}$

where the vector A_(B)=[1_(L) ^(T)0_(N/M−L) ^(T)1_(L) ^(T)0_(N/M−L) ^(T). . . 1_(L) ^(T)0_(N/M−L) ^(T)]_(1×N) ^(T). To individually estimate thecascaded channel of

-

_(k)-

, k=1,2, . . . , M , {circumflex over (v)} should first be transformedinto the FD. The

-

_(k)-

CSI can be extracted from the interleaved subcarriers by means of {tildeover (D)}_(F,k), and the FD estimates on the interleaved subcarriers arethen interpolated by {tilde over (L)}_(k). If TD channel estimation isrequired, F_(N) ^(H) is applied. Therefore,

-

_(k)-

can be estimated by:

{circumflex over (v)}_(k)=A_(L)F_(N) ^(H){tilde over (L)}_(k){tilde over(D)}_(F,k)F_(N){circumflex over (v)},

k=1,2, . . . , M.   (18)

where {tilde over (L)}_(k) represents an interpolation operator in theFD. Because the signal after {tilde over (D)}_(F,k) is considered asexpended channel samples in the FD, the signal would be interpolated inthe TD after the inverse DFT (IDFT) operation. Utilizing the channelestimates ĝ_(k) and {circumflex over (v)}_(k), h_(k) can be estimated asin equation (13).

In the CDM approach, the original source pilot must be replaced by theorthogonal source pilots. This process is similar to the TDM approachwithout the segmenting and reallocating steps, where the replaced pilotsmust be designed to be orthogonal to other pilots. The filtering matrixcan thus be reformulated as:

W _(k)

Φ(u _(S,k))F _(N) ^(H)Θ⁻¹(ũ _(S))F _(N), k=1,2, . . . , M.   (19)

FIG. 5 presents the slot structure of CDM approach in a 2-parallel-relaynetwork. By following the filtering matrix design in equation (19), theLS channel estimation on the

_(k)-

link can be achieved by:

$\begin{matrix}{{{\hat{g}}_{k} = {\frac{1}{N\sqrt{P_{S}}}A_{L}{\Phi^{H}( u_{k} )}y_{D}}},{k = 1},2,\ldots \mspace{14mu},{M.}} & (20)\end{matrix}$

The cascaded

-

_(k)-

CIR vector v_(k) can be estimated with:

$\begin{matrix}{{{\hat{v}}_{k} = {\frac{1}{N\sqrt{P_{S}}}A_{L}{\Phi^{H}( u_{S,k} )}y_{D}}},{k = 1},2,\ldots \mspace{14mu},{M.}} & (21)\end{matrix}$

Utilizing the channel estimates ĝ_(k) and {circumflex over (v)}_(k),h_(k) can be estimated as in equation (13). It can be proved that thefiltering matrices written in equations (9), (14) and (19) are in theform of a circulant matrix or the equivalent F_(N) ⁻¹Θ(v)F_(N) [39] andcan therefore be considered a TDL filter.

The BCRB of disintegrated channel estimation in single relay scenariohas previously been derived. However, multiple relay and filteringmatrix are considered into the signal model, and the property of theproposed filtering matrix should be taken into consideration. To extendthe derivation from single to multiple relay, a general form ofexponential integral function ε_(n)(•) has been introduced. As a result,the BCRB is included as a spacial case of our general BCRB derivation.The Fisher information matrix (FIM) can be expressed as:

$\begin{matrix}{ {J = {E_{z}\{ {E_{y_{D}|z}\{ {\frac{{\partial\ln}\; {p( {y_{D},z} )}}{\partial z^{*}}( \frac{{\partial\ln}\; {p( {y_{D},z} )}}{\partial z^{*}} )^{H}} z} \}}} \},} & (22)\end{matrix}$

where the parameter z consists of all channel pairs of

-

_(k) and

_(k)-

links, i.e.:

z=[z₁ ^(T)z₂ ^(T) . . . z_(M) ^(T)]^(T),   (23)

and each channel pair z_(k) contains both g_(k) and h_(k), i.e.,z_(k)=[g_(k) ^(T)h_(k) ^(T)]^(T). The FIM can be shown to be a blockdiagonal matrix, i.e., J=Θ([J₁J₂ . . . J_(M)]). Each diagonal submatrixof J can be expressed as:

$\begin{matrix}{{J_{k} = \begin{bmatrix}J_{k}^{(11)} & J_{k}^{(12)} \\J_{k}^{(21)} & J_{k}^{(22)}\end{bmatrix}},{k = 1},2,\ldots \mspace{14mu},M,} & (24)\end{matrix}$

where the elements of J_(k) can be derived as:

J _(k) ⁽¹¹⁾=α_(k) ²ρ_(h) _(k) ² c ₁ NP _(s) I+c ₁ NP _(k) I+Nσ _(k)⁴α_(k) ⁴ c ₃ I+R _(g) _(k) ⁻¹,

J _(k) ⁽²²⁾=α_(k) ² c ₂ NP _(s) I+R _(h) _(k) ⁻¹,

J _(k) ⁽¹²⁾ =J _(k) ⁽²¹⁾=0.   (25)

In equation (25), ρ_(h) _(k) ²∥h_(k)∥², σ_(k) ²=E{∥n_(k)∥²}, R_(g) _(k)=E{g_(k)g_(k) ^(H)}, R_(h) _(k) =E{h_(k)h_(k) ^(H)}; c₁, c₂ and c₃ areshown in equation (26):

$\begin{matrix}{\mspace{79mu} {{c_{1} = {\frac{1}{\sigma_{D}^{2}}{Be}^{B}{ɛ_{M}(B)}}},\mspace{79mu} {c_{2} = {\frac{\rho_{_{k}}^{2}}{\sigma_{D}^{2}}{Be}^{B}{ɛ_{M + 1}(B)}}},\mspace{79mu} {B = \frac{\sigma_{D}^{2}}{a_{k}^{2}\sigma_{k}^{2}\rho_{_{k}}^{2}}},{c_{3} = {\frac{B^{M}e^{B}}{\alpha_{k}^{2}\sigma_{D}^{2}\sigma_{k}^{2}{M!}}( {{( {- 1} )^{M}( {e^{- B} - {B\; {ɛ_{1}(B)}}} )} + {\sum\limits_{s = 0}^{M - 1}{\frac{{( {M - s - 1} )!}( {- 1} )^{s}}{B^{M - s - 1}}{ɛ_{M - s}(B)}}}} )}},}} & (26)\end{matrix}$

where ρ_(g) _(k) ²=∥g_(k)∥², σ_(D) ²=E{∥n_(D)∥^(2} and E) _(n)(•)denotes the general form of exponential integral function defined as

${ɛ_{n}(x)} = {\int_{1}^{\infty}{\frac{e^{- {xt}}}{t^{n}}{dt}}}$

and has the iterative property

${{ɛ_{n + 1}(x)} = {\frac{1}{n}( {e^{- x} - {x\; {ɛ_{n}(x)}}} )}},{n \geq 1.}$

The BCRBs of g_(k) and h_(k) can be obtained from the inverses of thediagonal submatrices of the FIM, i.e.:

BCRB _(g) _(k) =tr{J _(k) ⁽¹¹⁾⁻¹},

BCRB _(h) _(k) =tr{J _(k) ⁽²²⁾⁻¹},

k=1,2, . . . , M.   (27)

Based on equations (9), (14), and (19), the MSEs of the proposeddisintegrated channel estimation approaches are obtained.

The MSEs of the

_(k)-

channel estimations can be expressed as equations 28-30, as follows:

$\begin{matrix}{{{MSE}_{T;_{k}} = {{\frac{N_{cp}}{N}\rho_{_{k}}^{2}\eta_{k}^{- 1}} + {\frac{N_{cp}}{N_{r}}\eta_{D}^{- 1}}}},{{FF}\text{/}{TDM}\text{-}{SITS}}} & (28) \\{{{MSE}_{F;_{k}} = {\frac{N_{cp}}{N}( {{\frac{1}{M}{\sum\limits_{k = 1}^{M}{\rho_{_{k}}^{2}\eta_{k}^{- 1}}}} + \eta_{D}^{- 1}} )}},{{FF}\text{/}{FDM}\text{-}{SITS}}} & (29) \\{{{MSE}_{C;_{k}} = {\frac{N_{cp}}{N}( {{\sum\limits_{k = 1}^{M}{\rho_{_{k}}^{2}\eta_{k}^{- 1}}} + \eta_{D}^{- 1}} )}},{{FF}\text{/}{CDM}\text{-}{SITS}},} & (30)\end{matrix}$

where

$\eta_{k} = {{\frac{P_{s}}{\sigma_{k}^{2}}\mspace{14mu} {and}\mspace{14mu} \eta_{D}} = \frac{P_{s}}{\sigma_{D}^{2}}}$

are the SNRs at the kth relay and the destination, respectively.

The MSEs of the

-

_(k)-

channel estimations can be obtained from those of the

_(k)-

channel estimations, i.e.:

MSE_(T;v)=MSE_(T;g),

MSE _(F;v) =M×MSE _(F;g),

MSE_(C;v)=MSE_(C;g).   (31)

The general MSE expressions of the

_(k)-

channel estimations can be obtained as:

$\begin{matrix}{{{MSE}_{h} \geq {N_{cp}\frac{{\rho_{h_{k}}^{2}\sigma_{{\hat{}}_{k}}^{2}} + \sigma_{{\hat{v}}_{k}}^{2}}{\rho_{_{k}}^{2} + \sigma_{{\hat{}}_{k}}^{2}}}},} & (32)\end{matrix}$

where σ_(ĝ) _(k) ² and σ_({circumflex over (v)}) _(k) ² are thevariances of the estimation errors and e_(ĝ) _(k) ande_({circumflex over (v)}) _(k) , respectively.

Computer simulations were conducted to verify the analysis resultsderived herein and to reconfirm the improvements obtained using thesystem of the present disclosure. The system studied herein follows theLTE specifications, in which the DFT size is 256 for a 3 MHz bandwidthtransmission, the CP length is 18, the sampling rate is 3.84 MHz, thesample duration is 260 ns, and the CP duration is 4.69 μs. In thefollowing simulations, all links, including

-

_(k) and

_(k)-

, k=1,2, . . . , M , are simulated with independent TDL models, in whichthe tap spacing is commonly set as the sample duration. The TDLparameters are obtained from the reduced variability models in TableA1-19: the International Telecommunication Union (ITU-R) channel modelfor rural macrocell (RMa) test environments of a non-line of sight(NLoS) scenario. The RMa NLoS channel has a mean delay spread of 21.37ns, root-mean-square (rms) delay spread of 36.68 ns, and maximum excessdelay time of 220 ns, which roughly corresponds to a 2-tap TDL channelmodel in the studied system. Each tap is simulated as an independentcomplex Gaussian random variable. To avoid the inter-carrierinterference (ICI) effects caused by the random frequency modulationresulting from the relative movement, the simulations are all conductedin the context of block fading. Although the channels are considered tobe time-varying, all the tap-weighting coefficients remain unchangedwithin an OFDM symbol duration and are updated every transmission slot.The quadrature-phase-shift keying (QPSK) mapping with Gray coding isemployed. Four parallel relays are simulated and cooperated in ½ coderate STBC. The SNR at

_(k) is set to 20 dB to represent moderately noisy conditions.

FIGS. 6-7 show the normalized MSEs (NMSEs) for the

_(k)-

and the

-

_(k) channel estimations, respectively, obtained using the proposedapproaches with 4 parallel relays. For simplicity, xDM is used torepresent TDM, FDM and CDM approaches. The simulation results indicatethat the NMSEs of the

_(k)-

and

-

_(k) channel estimations obtained using the xDM approaches are close tothose obtained through the statistical derivations described inequations 28-32. The NMSEs of the

_(k)-

channel estimation obtained using the FDM and CDM are lower than thatobtained using the TDM, in which case η _(D) dominates the performance.In practice, the pilot with size N is an up-sampled version of the pilotwith size N_(r) in the channel estimation approach. Therefore, thefull-length pilot can benefit channel estimation accuracy by up-samplingthe CIR. As shown in equations 28-30, although the segmented pilot inTDM helps to avoid IRI, it unavoidably sacrifices the benefit gained byup-sampling by N_(r)/N times. Therefore, the FDM and the CDM can achievelower NMSEs. Comparing FDM with CDM, FDM only uses SITS in

_(k)-

channel estimation. The IRI obtained is reduced by a factor of 1/M. Thesimulation with η _(k)=20 dB shows that the IRI has a small effect onthe channel estimation approaches disclosed herein. The NMSEs obtainedusing the present approaches with LS channel estimations differ byseveral dBs from the BCRBs derived herein. This can also be observed inthe results of previous research on BCRBs, wherein an iterative methodwas also suggested. Sophisticated estimators other than the LS channelestimation, e.g., a minimum mean-square error (MMSE) channel estimation,may be exploited in conjunction with additional channel statistics toimprove the channel estimation performance.

The

-

_(k) channel estimation can be achieved by the deconvolution of the

-

_(k)-

and

_(k)-

CIRs in the TD. Meanwhile, the deconvolution process in the TD can beconducted by dividing the estimates of

-

_(k)-

by the estimates of

_(k)-

in the FD. Because the TD deconvolution or the FD division processinevitably increases the estimation error, the NMSEs of the

-

_(k) channel estimations are usually higher than those of

_(k)-

channel estimates. Meanwhile, the FDM spends the advantage earned fromits

-

_(k) channel estimation and

-

_(k)-

channel estimation, as the FD channel reconstruction employed in the FDMresults in a non-negligible interpolation error.

The LS channel estimation is employed in the

-

_(k) channel estimation to allow a fair comparison. Prior studies haveshown that a 5 bit/dim quantization process can provide sufficientaccuracy for the CF strategy. Without loss of generality and for thesimplicity of channel coding, CFs with the 6 bit/dim scheme weresimulated. Considering both the in-phase and quadrature-phase componentsof the CIRs, 12 bit/tap was chosen for the simulations. Furthermore,twice and four times the channel standard deviation ρ_(g) _(k) were setas the maximum quantization range. The difference between two contiguousquantization levels is uniformly set to be Δ=0.125ρ_(g) _(k) , 0.25ρ_(g)_(k) . Therefore, the quantization noise σ_(q) _(e) ², is 0.0013ρ_(g)_(k) ², 0.0052ρ_(g) _(k) ². A tradeoff apparently exists between themaximum quantization range and the quantization noise. For a higherquantization range, a wide range of channel estimates can be sent to thedestination, but the variance of the quantization error unavoidablyincreases if the number of quantization levels is kept unchanged. Inaddition, a simple (7,4) cyclic block coding is employed to reduce biterror propagation. As shown in FIG. 6, the NMSEs of the

-

_(k) channel estimation after the channel reconstruction conducted inthe destination for the CF strategy are much higher than those obtainedusing the proposed techniques.

FIG. 8 plots the SER versus the average SNR at a destination obtainedusing the proposed technique with STBC when the average SNR η _(k) atthe relay nodes is 20 dB. As η _(D) increases from 0 to 30 dB, the errorfloor begins at approximately η _(D)=η _(k), and η _(D)>η _(k)determines the interference-dominant region. This finding shows that theSER performance of the STBC is predominantly determined by η _(k)regardless of any increase of η _(D) in the interference-dominantregion; meanwhile, increasing η _(k) cannot reduce the SERssignificantly. It can also be observed that the proposed xDM approachescan yield much lower SERs than the CF strategy, even if the CF relayingstrategy takes advantage of some basic channel coding. This is mainlybecause estimation errors are unavoidable in the

-

_(k) channel estimations due to the relay noises, quantization errorsare unavoidable in the quantizer at the relay nodes and error alsooccurs within the signal detection conducted at the destination due tothe non-negligible destination noise. An error correction code can helpreduce the third impairment but cannot effectively reduce the first two.The SER obtained using the CF relaying strategy can be reduced to thoseof xDM approaches by increasing n _(D) by approximately 20 dB. In otherwords, the destination noise is the dominant impairment of the threeabove-mentioned errors in the CF relaying strategy. An ideal DF relayingwith STBC is also simulated here. It was also assumed that the signalsvia

-

links are perfectly demodulated and decoded in relays. Then, the relayscooperatively establish D-STBC in the destination.

In summary, the three approaches disclosed herein can successfullyperform the disintegrated channel estimation. From the figures, it canbe seen that the BCRBs can be considered effective performancebenchmarks because the NMSEs obtained using the proposed approaches areclose to the derived BCRBs. This finding indicates that the proposedchannel estimation approaches can effectively coordinate IRI to achievelow NMSE and thus low SER, as shown in FIG. 8. Even with no assistancefrom error correction coding, the SER can be as low as 10⁻² to 10⁻⁴ fora moderate SNR at the destination η _(D)=20 dB. From the MSE derivationand SER simulation, it can be observed that when η _(D)<η _(k), η _(D)dominates the SER performance. Therefore, as η _(D) increases, the MSEsand SERs can be reduced. When η _(D)>η _(k), n _(k) dominates the SERperformance. Comparing the structures of the proposed estimators, theFDM approach has higher spectral and energy efficiencies because noadditional CP is required to avoid IBI and the number of relays limitthe filtering matrix.

Because the complexities of achieving the STBCs based on the FF or CFrelaying strategies are similar, the complexity comparison may focus onthese strategies and the channel estimation processes. The proposedchannel estimation approaches are all based on a similar relay filteringprocess W_(k). Because an FFT operation requires N log₂N multiplicationsand a diagonal matrix requires N multiplications, W_(k) requires 2(Nlog₂N+N_(r) log₂ N_(r))+N+N_(r) multiplications and 2(N log₂N+N_(r) log₂N_(r))+N accumulations for the TDM, 2N log₂N multiplications and 2Nlog₂N+N accumulations for the FDM, and 2N log₂ N+N multiplications and2N log₂N+N accumulations for the CDM. The filtering process W_(k) of xDMapproaches can be pre-calculated and pre-stored on the kth relay afteru_(s) is assigned because all the other components in W_(k) areconstant. In each training cycle, an xDM strategy requires N²multiplications and N²+N accumulations, in which N accumulations resultfrom superimposing pilots. Considering the channel estimation at

, TDM requires 2N_(r) log₂ N+N_(r) multiplications, and both CDM and FDMrequire 2N log₂ N+N multiplications. For a disintegrated channelestimation in a CF relaying network, the FD LS channel estimationrequires at least 2N log₂ N+N multiplications and 2Nlog₂N accumulationsper training cycle. Although a CF relay seems to have lower complexitythan xDM approaches on a relay node, the complexity of the quantization,compression and forwarding of the source-relay CIR to the destinationnode has not been taken into account. The xDM strategy employs theoriginal pilot or subcarriers employed by the conventional AF and DFrelaying strategies, keeping the protocol unchanged, and therefore hashigher relay network scalability and feasibility. However, CF needs toforward the source-relay CIR estimates to the destination; therefore,extra bandwidth is needed. The extra bandwidth is significant;meanwhile, the complexities of quantization, compression andremodulation of the source-relay CIR estimates are also high, even ifthe protocol can be adjusted accordingly (and/or in a timely manner).

Additional graphs illustrating simulated performance tests in connectionwith the system of the present disclosure are illustrated in FIGS. 9-12.

FIG. 9 illustrates the SER versus the average SNR at the destinationobtained using the system of the present disclosure with STBC, when theaverage SNR at the relays n_⁻R is 10, 20 and 60 dB. The SERs obtainedusing the perfect CE with STBC and the conventional AF technique withoutSTBC are also plotted as comparison baselines. The SERs obtained usingFF/CDM-ST with 4 relays are also shown in this figure for comparison.Three things can be observed from FIG. 9: (1) the SERs obtained usingthe proposed techniques are close to those obtained using the perfect CEon any relay SNR; (2) when the relay SNR is high (≧20 dB), the proposedtechnique with STBC outperforms the conventional AF and approaches theperfect CE; and (3) when the relay SNR is low (≦10 dB), the CE errors ofthe proposed methods degrade the performance of the MRC at thedestination. Therefore, the system of the present disclosure and theperfect CE can slightly outperform the conventional AF technique. Thissimulation indicates that spatial diversity can be achieved withaccurate disintegrated CE in relay networks, and that the SERperformance is significantly affected by the relay SNR.

The MSEs of LS CE on R_k-D are shown in FIG. 10. It may be noted thatthe FF/FDM-ST produces lower estimation errors than FF/CDM-ST andFF/TDM-ST. This is because FF/FDM-ST is implicitly involved with a FDtruncator, which functions as an ideal lowpass filter (LPF) suppressingthe out-of-band noise. The disadvantage of FF/FDM-ST is that thefiltering matrix utilizes more spectrum resources; leading to a stricterlimit on the number of relays. Another phenomenon in this simulationshows that if the relay noise grows to a certain level, the crossoversof the MSE curves occur when the relays and the destination have thesame noise power. This result indicates that the FF/FDM-ST and FF/CDM-STtechniques have less tolerance to the relay noise but high tolerance todestination noise.

The MSEs of S-R_k-D CE are shown in FIG. 11. The MSEs obtained using theFF/FDM-ST increase slightly, because the FD channel reconstructionemployed in the FF/FDM-ST results in a non-negligible interpolationerror. The MSEs of S-R_k CE are shown in FIG. 12. The LS S-R_k CE can beobtained through the deconvolution process in the TD or through thedivision of the S-R_k-D CTF to the R_k-D CTF. The division processresults in a noise enhancement problem, which represents the mainchallenge in deriving a closed-form MSE formulation. The S-R_k CEexhibits a worse performance than the R_k-D and S-R_k-D CE. FIG. 12 alsoillustrates that the noise enhancement problem increases the MSEs byapproximately one half to one order of magnitude.

FIG. 13 is a diagram illustrating hardware components capable ofimplementing the system of the present disclosure The system of thepresent disclosure can be implemented in the relay nodes discussed abovein connection with FIG. 2, as well as the destination node discussedtherein. The hardware components, indicated generally at 50, couldinclude an antenna system 52 for receiving and transmitting signals, aradiofrequency (RF) transceiver 54, and a processor 56 controlling thetransceiver 54. The transceiver 54 and antenna 52 could permitcommunications using OFDM protocols, and/or could support 3G, 3GPP, 4G,LTE, 5G, and other wireless communication protocols, as well as otherprotocols. The processor 56 could be a suitable microprocessor,microcontroller, digital signal procesor (DSP), application-specificintegratd circuit (ASIC), field-programmable gate array (FPGA), etc. Theprocessing steps disclosed herein could be embodied as a channelestimation module 58 executed by the processor 56, e.g.,computer-readable instructions executable by the processor 56. Thehardware components 50 could form part of base station equipment, relaynode eqiupment, and/or mobile device (e.g., cellular telephone)equipment.

A generic FF relaying strategy with multiplexed superimposed trainingsequences has been disclosed herein. The generalized filteringapproaches effectively multiplex the superimposed training sequences toovercome IRI. Moreover, distinguished channel estimation techniquesbased on the generalized filtering approaches have been systematicallyderived, statistically analyzed and compared with the BCRBs.

The following additional equations and derivations are provided forreference.

The FIM in equation (22) can be reformulated as equation (33):

$\begin{matrix}{J = {E_{z}{\{ {E_{y_{D}z}\{ {{\frac{{\partial\ln}\; {p( {y_{D}z} )}}{\partial z^{*}}( \frac{{\partial\ln}\; {p( {y_{D}z} )}}{\partial z^{*}} )^{H}} + {\frac{{\partial\ln}\; {p(z)}}{\partial z^{*}}( \frac{{\partial\ln}\; {p(z)}}{\partial z^{*}} )^{H}}} \}} \}.}}} & (33)\end{matrix}$

By considering the parameter vector z defined in (23), a matrix F can bedefined as

$\begin{matrix}\begin{matrix}{F = {E_{z}\{ {\frac{{\partial\ln}\; {p( {y_{D}z} )}}{\partial z^{*}}( \frac{{\partial\ln}\; {p( {y_{D}z} )}}{\partial z^{*}} )^{H}} \}}} \\{{= \begin{bmatrix}F_{(11)} & F_{(21)} & \ldots & F_{({M\; 1})} \\F_{(21)} & F_{(21)} & \; & \vdots \\\vdots & \; & \ddots & \vdots \\F_{({M\; 1})} & \ldots & \ldots & F_{({MM})}\end{bmatrix}},}\end{matrix} & (34) \\{Where} & \; \\\begin{matrix}{F_{({km})} = {E_{y_{D}z}\{ {\frac{{\partial\ln}\; {p( {y_{D}z} )}}{\partial z_{k}^{*}}( \frac{{\partial\ln}\; {p( {y_{D}z} )}}{\partial z_{m}^{*}} )^{H}} \}}} \\{= {\begin{bmatrix}F_{({km})}^{11} & F_{({km})}^{12} \\F_{({km})}^{21} & F_{({km})}^{22}\end{bmatrix}.}}\end{matrix} & (35)\end{matrix}$

from the signal model in (7), the likelihood function can be expressedas equation (36):

$\begin{matrix}{{p( {y_{D}z} )} = {\frac{1}{\pi^{N}{R_{nz}}}{\exp( {{- ( {y_{D} - ( {{\sum\limits_{k = 1}^{M}{\sqrt{P_{s}}\alpha_{k}{\Phi ( g_{k} )}{\Phi ( h_{k} )}u_{S,k}}} + {\sum\limits_{k = 1}^{M}{\sqrt{P_{k}}{\Phi ( g_{k} )}u_{k}}}} )} )^{H}}R_{nz}^{- 1}{ \quad( {y_{D} - ( {{\sum\limits_{k = 1}^{M}{\sqrt{P_{s}}\alpha_{k}{\Phi ( g_{k} )}{\Phi ( h_{k} )}u_{S,k}}} + {\sum\limits_{k = 1}^{M}{\sqrt{P_{k}}~{\Phi ( g_{k} )}u_{k}}}} )} ) ).}} }}} & (36)\end{matrix}$

The mean and covariance matrix of y_(D) are obtained as:

$\begin{matrix}{{\mu = {{\sum\limits_{k = 1}^{M}{\sqrt{P_{s}}\alpha_{k}{\Phi ( g_{k} )}{\Phi ( h_{k} )}u_{S,k}}} + {\sum\limits_{k = 1}^{M}{\sqrt{P_{k}}{\Phi ( g_{k} )}u_{k}}}}}\begin{matrix}{R_{nz} = {{\sum\limits_{k = 1}^{M}{\alpha_{k}^{2}\sigma_{k}^{2}{\Phi ( g_{k} )}{\Phi^{H}( g_{k} )}}} + {\sigma_{D}^{2}I}}} \\{= {F^{H}\{ {\Theta ( \lbrack {{\sum\limits_{k = 1}^{M}{\alpha_{k}^{2}\sigma_{k}^{2}{{\overset{\sim}{}}_{k,i}}^{2}}} + \sigma_{D}^{2}} \rbrack_{i = 1}^{N} )} \} {F.}}}\end{matrix}} & (37)\end{matrix}$

To derive a closed form from equation (36), the expectations in equation(38) must be evaluated:

$\begin{matrix}{\mspace{79mu} {{{E_{z}\{ {{\Phi^{H}( h_{k} )}{\Phi ( h_{k} )}} \}} = {\rho_{h_{k}}^{2}I}}\begin{matrix}{\mspace{79mu} {{E_{z}\{ R_{nz}^{- 1} \}} = {E_{z}\{ ( {{\sum\limits_{k = 1}^{M}{\alpha_{k}^{2}\sigma_{k}^{2}{\Phi ( g_{k} )}{\Phi^{H}( g_{k} )}}} + {\sigma_{D}^{2}I}} )^{- 1} \}}}} \\{= {{F^{H}( {\Theta ( \lbrack {E_{z}\{ \frac{1}{{\sum\limits_{k = 1}^{M}{\alpha_{k}^{2}\sigma_{k}^{2}{{\overset{\sim}{}}_{k,i}}^{2}}} + \sigma_{D}^{2}} \}} \rbrack_{i = 1}^{N} )} )}F}} \\{= {\frac{1}{\alpha_{k}^{2}\sigma_{k}^{2}\rho_{_{k}}^{2}}{\exp ( \frac{\sigma_{D}^{2}}{\alpha_{k}^{2}\sigma_{k}^{2}\rho_{_{k}}^{2}} )}{ɛ_{M}( \frac{\sigma_{D}^{2}}{\alpha_{k}^{2}\sigma_{k}^{2}\rho_{_{k}}^{2}} )}I}} \\{= {\frac{1}{\sigma_{D}^{2}}{Be}^{B}{ɛ_{M}(B)}I}} \\{= {c_{1}I}}\end{matrix}\mspace{79mu} {{E_{z}\{ {R_{nz}^{- 1}{\Phi ( g_{k} )}} \}} = 0}\mspace{76mu} {{E_{z}\{ {{\Phi ( g_{k} )}R_{nz}^{- 1}{\Phi^{H}( g_{k} )}} \}} = {{\frac{\rho_{_{k}}^{2}}{\sigma_{D}^{2}}{Be}^{B}{ɛ_{M + 1}(B)}I} = {c_{2}I}}}{{E_{z}\{ {{\Phi ( g_{k} )}R_{nz}^{- 2}{\Phi^{H}( g_{k} )}} \}} = {{\frac{B^{M}e^{B}}{\alpha_{k}^{2}\sigma_{D}^{2}\sigma_{k}^{2}}{\int_{B}^{\infty}{\frac{1}{x^{M}}{ɛ_{M + 1}(x)}{dx}}}} = {{\quad\quad}\frac{B^{M}e^{B}}{\alpha_{k}^{2}\sigma_{D}^{2}\sigma_{k}^{2}{M!}} {\quad\quad}( {{( {- 1} )^{M}( {e^{- B} - {B\; {ɛ_{1}(B)}}} )} + {\sum\limits_{s = 0}^{M - 1}{\frac{{( {M - s - 1} )!}( {- 1} )^{s}}{B^{M - s - 1}}{ɛ_{M - s}(B)}}}} ){I.}}}}}} & (38)\end{matrix}$

From equation (20), by assuming P_(s)=P_(k), the LS channel estimationof the ĝ_(k) can be evaluated using equations (7) and (8), i.e.,

${\hat{g}}_{k} = {g_{k} + {\frac{1}{N\sqrt{P_{s}}}A_{L}{\Phi^{H}( u_{k} )}{( {{\sum\limits_{k = 1}^{M}( {{\Phi ( g_{k} )}W_{k}n_{k}^{\prime}} )} + n_{D}} ).}}}$

The second term of the above equation represents the estimation error ofĝ_(k) and can be rewritten as:

$\begin{matrix}{e_{{\hat{}}_{k}} = {\frac{1}{N\sqrt{P_{S}}}A_{L}{\Phi^{H}( u_{k} )}{( {{\sum\limits_{k = 1}^{M}( {{\Phi ( g_{k} )}W_{k}{n^{\prime}}_{k}} )} + n_{D}} ).}}} & (39)\end{matrix}$

Considering the generic form of the FF/CDM-SITS filtering matrix on thekth relay, the following should be taken into account:

W_(k)W_(k) ^(H)=I_(N),   (1)

E{n′_(k)n′_(k) ^(H)}=σ_(k) ²I_(N),   (2)

E{Φ(g _(k))Φg _(k) ^(H}) =M(g _(k))=ρ_(g) _(k) ² I _(N),   (3)

Φ(u _(k))^(H)Φ(u _(k))=NI _(N),   (4)

E{n_(D)n_(D) ^(H)}=σ_(D) ²I_(N),   (5)

A_(L)A_(L) ^(H)=I_(N) _(cp) ,   (6)

The covariance matrix of ê_(ĝ) _(k) can be derived as:

$\begin{matrix}\begin{matrix}{{M( e_{{\hat{g}}_{k}} )} = {E\{ {e_{{\hat{g}}_{k}}e_{{\hat{g}}_{k}}^{H}} \}}} \\{= {{\frac{1}{N^{2}P_{s}}{\sum\limits_{k = 1}^{M}\; {\sigma_{k}^{2}\{ {A_{L}{\Phi^{H}( u_{k} )}{M( g_{k} )}{\Phi ( u_{k} )}A_{L}^{H}} \}}}} +}} \\{{\frac{1}{N^{2}}\sigma_{D}^{2}\{ {A_{L}{\Phi^{H}( u_{k} )}{\Phi ( u_{k} )}A_{L}^{H}} \}}} \\{= {\frac{1}{{NP}_{s}}{\{ {{\sum\limits_{k = 1}^{M}\; {\rho_{g_{k}}^{2}\sigma_{k}^{2}I_{L}}} + {\sigma_{D}^{2}I_{L}}} \}.}}}\end{matrix} & (40)\end{matrix}$

The MSE can be evaluated with MSE_(C;g) _(k) =tr{M(e_(ĝ) _(k) )}, wheretr{•} denotes the trace of its matrix argument.

The channel estimation error of

-

-

can be expressed as:

$\begin{matrix}{e_{{\hat{v}}_{k}} = {\frac{1}{N\sqrt{P_{s}}}A_{L}{\Phi^{H}( u_{S,k} )}{( {{\sum\limits_{k = 1}^{M}\; ( {{\Phi ( g_{k} )}W_{k}{n^{\prime}}_{k}} )} + n_{D}} ).}}} & (41)\end{matrix}$

Therefore, the covariance matrix of e_({circumflex over (v)}) _(k) canbe derived as:

$\mspace{760mu} {(42)\begin{matrix}{{M( e_{{\hat{v}}_{k}} )} = {E\{ {e_{{\hat{v}}_{k}}e_{{\hat{v}}_{k}}^{H}} \}}} \\{= {{\frac{1}{N^{2}P_{s}}{\sum\limits_{k = 1}^{M}{E\{ {A_{L}{\Phi^{H}( u_{S,k} )}{\Phi ( g_{k} )}W_{k}{n^{\prime}}_{k}{n^{\prime}}_{k}^{H}W_{k}^{H}{\Phi^{H}( g_{k} )}{\Phi ( u_{S,k} )}A_{L}^{H}} \}}}} +}} \\{{\frac{1}{N^{2}P_{s}}E\{ {A_{L}{\Phi^{H}( u_{S,k} )}n_{D}n_{D}^{H}{\Phi ( u_{S,k} )}A_{L}^{H}} \}}} \\{= {\frac{1}{{NP}_{s}}{\{ {{\sum\limits_{k = 1}^{M}{\rho_{g_{k}}^{2}\sigma_{k}^{2}I_{L}}} + {\sigma_{D}^{2}I_{L}}} \}.}}}\end{matrix}}$

The MSE can thus be evaluated asMSE_(C;v)=tr{M(e_({circumflex over (v)}) _(k) )}.

The MSE of the

-

_(k) channel estimation is derived in the FD for simplicity. Theestimates of g_(k) and v_(k) can be expressed as:

{tilde over (ĝ)} _(k) ={tilde over (g)} _(k) +e _({tilde over (ŷ)}) _(k)and {tilde over ({circumflex over (v)})} _(k) ={tilde over (v)} _(k) +e_({tilde over ({circumflex over (v)})}) _(k) ,   (43)

where e_({tilde over (ĝ)}) _(k) ande_({tilde over ({circumflex over (v)})}) _(k) are the errors occurringwith the {tilde over (g)}_(k) and {tilde over (v)}_(k) channelestimates, respectively. From equation (13), the estimates {tilde over(ĝ)}_(k) and {tilde over ({circumflex over (v)})}_(k) can be obtained bythe FD LS channel estimation. Meanwhile, the TD channel estimation canbe conducted, and its error can be expressed as:

                                          (44) $\begin{matrix}{{{\hat{h}}_{k} = {h_{k} + {{F_{N}^{H}( {{{\Theta^{- 1}( {{\overset{\sim}{g}}_{k} + e_{{\hat{\overset{\sim}{g}}}_{k}}} )}{\Theta ( {\overset{\sim}{g}}_{k} )}} - I} )}F_{N}h_{k}} + {F_{N}^{H}{\Theta^{- 1}( {{\overset{\sim}{g}}_{k} + e_{{\hat{\overset{\sim}{g}}}_{k}}} )}F_{N}e_{{\overset{\sim}{v}}_{k}}}}}\mspace{79mu} {e_{h_{k}} = {{{\Phi ( \frac{- e_{{\hat{g}}_{k}}}{g_{k} + e_{{\hat{g}}_{k}}} )}h_{k}} + {{\Phi ( \frac{1}{g_{k} + e_{{\hat{g}}_{k}}} )}{e_{{\hat{v}}_{k}}.}}}}} & \;\end{matrix}$

The division in the above equation is an element-wise division. The

-

_(k) channel can be derived from the

-

_(k)-

and

_(k)-

channel estimations as:

{tilde over (ĥ)} _(k) =h _(k) +F _(N) ^(H)[Θ⁻¹({tilde over (g)} _(k) +e_({tilde over (ĝ)}) _(k) )Θ({tilde over (g)} _(k))−I ]F _(N) h _(k) +F_(N) ^(H)Θ⁻¹({tilde over (g)} _(k) +e _({tilde over (ĝ)}) _(k) )F _(N) e_({circumflex over (v)}) _(k) .   (45)

Therefore, the channel estimation error of {tilde over (ĥ)}_(k) is aswritten in equation (32), and the covariance matrix is derived as:

$\begin{matrix}\begin{matrix}{{M( e_{{\hat{h}}_{k}} )} = {E\{ {e_{{\hat{h}}_{k}}e_{{\hat{h}}_{k}}^{H}} \}}} \\{= {E\{ {{{\Phi ( \frac{- e_{{\hat{g}}_{k}}}{g_{k} + e_{{\hat{g}}_{k}}} )}h_{k}h_{k}^{H}{\Phi^{H}( \frac{- e_{{\hat{g}}_{k}}}{g_{k} + e_{{\hat{g}}_{k}}} )}} + \quad} }} \\ {\Phi ( \frac{1}{g_{k} + e_{{\hat{g}}_{k}}} )e_{{\hat{v}}_{k}}e_{{\hat{v}}_{k}}^{H}{\Phi^{H}( \frac{1}{g_{k} + e_{{\hat{g}}_{k}}} )}} \} \\{\geq {{\rho_{h_{k}}^{2}\frac{\sigma_{{\hat{g}}_{k}}^{2}}{\rho_{g_{k}}^{2} + \sigma_{{\hat{g}}_{k}}^{2}}I_{L}} + {\sigma_{{\hat{v}}_{k}}^{2}\frac{1}{\rho_{g_{k}}^{2} + \sigma_{{\hat{g}}_{k}}^{2}}I_{L}}}} \\{{= {\frac{{\rho_{h_{k}}^{2}\sigma_{{\hat{g}}_{k}}^{2}} + \sigma_{{\hat{v}}_{k}}^{2}}{\rho_{g_{k}}^{2} + \sigma_{{\hat{g}}_{k}}^{2}}I_{L}}},}\end{matrix} & (46)\end{matrix}$

where Jensen's inequality is applied. Therefore, the MSE of the

-

_(k) channel estimation can be evaluated with MSE_(C;h) _(k) =tr{M(e_(ĥ)_(k) )}. The MSEs of the

-

_(k) channel estimations in the proposed TDM and FDM approaches can bederived by substituting the error variances of

-

_(k)-

and

_(k)-

channel estimation into equation (46).

Based on the signal model in equation (7), the relay matrix of theproposed TDM approach in equation (9) and the

_(k)-

channel estimation in equation (11), the channel estimation error can beevaluated as:

$\begin{matrix}{e_{{\hat{g}}_{k}} = {\frac{1}{N_{r}\sqrt{P_{s}}}A_{L,N_{r}}{\Phi^{H}( {D_{T,k}u_{k}} )}{\{ {{{\Phi ( {A_{L,N_{r}}g_{k}} )}D_{T,k}W_{k}{n^{\prime}}_{k}} + {D_{T,k}{n_{D}.}}} \}.}}} & (47)\end{matrix}$

According to the following:

${(1)\mspace{20mu} D_{T,k}D_{T,m}^{H}} = \{ {\begin{matrix}{I,} & {k = m} \\{0_{N},} & {k \neq m}\end{matrix},{{(2)\mspace{20mu} W_{k}W_{k}^{H}} = {\frac{N_{r}}{N}D_{T,k}^{H}D_{T,k}}},{{(3)\mspace{20mu} {\Phi ( {A_{N,}g_{k}} )}{\Phi^{H}( {A_{N,}g_{k}} )}} = {{p_{gk}^{2}{I_{N_{r},}(4)}\mspace{20mu} {\Phi ( {D_{T,k}u_{S,k}} )}{\Phi^{H}( {D_{T,k}u_{S,k}} )}} = {N_{r}I_{N_{r}}}}},{{{(5)\mspace{20mu} A_{L,N_{r}}A_{L,N_{r}}^{H}} = I_{L}};}} $

the covariance matrix is derived as:

$\begin{matrix}\begin{matrix}{{M( e_{{\hat{g}}_{k}} )} = {{\frac{1}{N_{r}^{2}P_{s}}\frac{N_{r}}{N}N_{r}\rho_{g_{k}}^{2}\sigma_{k}^{2}I_{L}} + {\frac{1}{N_{r}^{2}P_{s}}N_{r}\sigma_{D}^{2}I_{L}}}} \\{= {{\frac{1}{{NP}_{s}}\rho_{g_{k}}^{2}\sigma_{k}^{2}I_{L}} + {\frac{1}{N_{r}P_{s}}\sigma_{D}^{2}{I_{L}.}}}}\end{matrix} & (48)\end{matrix}$

The MSE can be evaluated as MSE_(T;g)=tr{M(e_(ĝ) _(k) )}.

From the

_(k)-

channel estimation in equation (12), the channel estimation error can beevaluated as:

$\begin{matrix}{e_{{\hat{v}}_{k}} = {\frac{1}{N_{r}P_{s}}A_{L,N_{r}}{\Phi^{H}( {D_{T,k}u_{S,k}} )}\{ {{{\Phi ( {A_{N_{r}}g_{k}} )}D_{T,k}W_{k}{n^{\prime}}_{k}} + {D_{T,k}n_{D}}} \}}} & (49)\end{matrix}$

and the covariance matrix can be written as:

$\begin{matrix}\begin{matrix}{{M( e_{{\hat{v}}_{k}} )} = {{\frac{1}{N_{r}^{2}P_{s}}\frac{N_{r}}{N}N_{r}\rho_{g_{k}}^{2}\sigma_{k}^{2}I_{L}} + {\frac{1}{N_{r}^{2}P_{s}}N_{r}\sigma_{D}^{2}I_{L}}}} \\{= {{\frac{1}{{NP}_{s}}\rho_{g_{k}}^{2}\sigma_{k}^{2}I_{L}} + {\frac{1}{N_{r}P_{s}}\sigma_{D}^{2}{I_{L}.}}}}\end{matrix} & (50)\end{matrix}$

The MSE can be evaluated as MSE_(T;v)=tr{M(e_({circumflex over (v)})_(k) )}.

Based on the signal model in equation (7), the relay matrix of theproposed FDM approach in equation (14) and the

_(k)-

channel estimation in equation (16), the channel estimation error can beevaluated as:

$\begin{matrix}{e_{{\hat{g}}_{k}} = {{\frac{1}{N\sqrt{P_{s}}}A_{L}{\Phi^{H}( u_{k} )}{\sum\limits_{k = 1}^{M}\; ( {{\Phi ( g_{k} )}W_{k}{n^{\prime}}_{k}} )}} + {\frac{1}{N\sqrt{P_{s}}}A_{L}{\Phi^{H}( u_{k} )}{n_{D}.}}}} & (51)\end{matrix}$

According to the following facts,

${{(1)\mspace{20mu} W_{k}W_{k}^{H}} = {{F_{N}^{H}D_{F,k}F_{N}} = W_{k}}},{{(2)\mspace{20mu} E\{ {{n^{\prime}}_{k}{n^{\prime}}_{k}^{H}} \}} = {\sigma_{k}^{2}I_{N}}},{{(3)\mspace{20mu} E\{ {{\Phi ( g_{k} )}{\Phi^{H}( g_{k} )}} \}} = {{M( g_{k} )} = {\rho_{g_{k}}^{2}I_{N}}}},{{(4)\mspace{20mu} {\Phi^{H}( u_{k} )}{\Phi ( u_{k} )}} = {NI}_{N}},{{(5)\mspace{20mu} E\{ {n_{D}n_{D}^{H}} \}} = {\sigma_{D}^{2}I_{N}}},{{(6)\mspace{20mu} A_{L}A_{L}^{H}} = I_{L}},{{(7)\mspace{20mu} A_{L}W_{k}A_{L}^{H}} = {\frac{1}{M}I_{L}}},$

the covariance matrix of e_(ĝ) _(k) , can be derived as:

                                          (52) $\begin{matrix}\begin{matrix}{{M( e_{{\hat{g}}_{k}} )} = {E\{ {e_{{\hat{g}}_{k}}e_{{\hat{g}}_{k}}^{H}} \}}} \\{= {{\frac{1}{N^{2}P_{s}}{\sum\limits_{k = 1}^{M}{E\{ {A_{L}{\Phi^{H}( u_{k} )}{\Phi ( g_{k} )}W_{k}{n^{\prime}}_{k}{n^{\prime}}_{k}^{H}W_{k}^{H}{\Phi^{H}( g_{k} )}{\Phi ( u_{k} )}A_{L}^{H}} \}}}} +}} \\{{\frac{1}{N^{2}P_{s}}E\{ {A_{L}{\Phi^{H}( u_{k} )}n_{D}n_{D}^{H}{\Phi ( u_{k} )}A_{L}^{H}} \}}} \\{= {\frac{1}{{NP}_{s}}{\{ {{\frac{1}{M}{\sum\limits_{k = 1}^{M}{\rho_{g_{k}}^{2}\sigma_{k}^{2}I_{L}}}} + {\sigma_{D}^{2}I_{L}}} \}.}}}\end{matrix} & \;\end{matrix}$

The MSE can then be evaluated as MSE_(F;g) _(k) =tr{M(e_(ĝ) _(k) )}.

From the

_(k)-

channel estimation in equation (17), the channel estimation error can beevaluated as:

$\begin{matrix}{e_{{\hat{v}}_{k}} = {\frac{1}{N\sqrt{P_{s}}}A_{L}F_{N}^{H}{\overset{\sim}{L}}_{k}{\overset{\sim}{D}}_{F,k}F_{N}{\Theta ( A_{B} )}{\Phi^{H}( u_{S} )}{\{ {{\sum\limits_{k = 1}^{M}\; ( {{\Phi ( g_{k} )}n_{k}} )} + n_{D}} \}.}}} & (53)\end{matrix}$

Let A=A_(L)F_(N) ^(H){tilde over (L)}_(k){tilde over(D)}_(F,k)F_(N)A_(B), then AA^(H)=MI_(L). Assuming σ_(k) ²=σ_(κ) ² andρ_(g) _(k) =σ_(g) _(κ) ², the covariance matrix of {circumflex over(v)}_(k) can be derived as:

$\begin{matrix}\begin{matrix}{{M( e_{{\hat{v}}_{k}} )} = {{\frac{1}{N^{2}P_{s}}\sigma_{k}^{2}{{A\Phi}^{H}( u_{S} )}{M( g_{k} )}{\Phi ( u_{S} )}A^{H}} + {\frac{1}{{NP}_{s}}\sigma_{D}^{2}I_{N_{cp}}}}} \\{= {\frac{1}{{NP}_{s}}{\{ {{\rho_{g_{k}}^{2}\sigma_{k}^{2}I_{L}} + {\sigma_{D}^{2}I_{L}}} \}.}}}\end{matrix} & (54)\end{matrix}$

The MSE can be evaluated as MSE_(F;v)=tr{M(e_({circumflex over (v)})_(k) )}.

Having thus described the invention in detail, it is to be understoodthat the foregoing description is not intended to limit the scope of thepresent invention. What is intended to be protected by Letters Patent isset forth in the following claims.

What is claimed is:
 1. A system for disintegrated channel estimation ina wireless communication network, comprising: a plurality of wirelessrelay nodes in communication with a source node and a destination node,each of said plurality of wireless relay nodes including a processor,wherien the processors cause the plurality of wireless relay nodes toeach: receive a pilot signal and a data signal from the source node;filter the pilot signal to generate a filtered pilot signal; process thefiltered pilot signal to generate an orthogonal pilot signal;superimpose the orthogonal pilot signal onto the filtered pilot signalto generate a superimposed filtered pilot signal; and transmit thesuperimposed filtered pilot signal to the destination node, wherein thedestination node receives the superimposed filtered pilot signals fromthe plurality of wireless realy nodes, and estimates disintegratedchannel information using the superimposed filtered pilot signals. 2.The system of claim 1, wherein the processors causes the plurality ofrelay nodes to each process the data signal into a space-time coded datasignal.
 3. The system of claim 2, wherein the processors cause theplurality of relay nodes to each transmit the space-time coded datasignal to the destination node.
 4. The system of claim 3, wherein thedestination node receives the space-time coded datadata signal andprocesses the space-time coded data signal using the disintegratedchannel information.
 5. The system of claim 1, wherein the plurality ofrelay nodes each operate in a filter-and-forward mode of operation. 6.The system of claim 5, wherein the plurality of relay nodes communicatewith the source node and the destination node using time-divisionmultiplexing (TDM).
 7. The system of claim 5, wherein the plurality ofrelay nodes communicate with the source node and the destination nodeusing frequency-division multiplexing (FDM).
 8. The system of claim 5,wherein the plurality of relay nodes communicate with the source nodeand the destination node using code-division multiplexing (CDM).
 9. Thesystem of claim 1, wherein the processors cause the pluarlity of relaynodes to perform filtering to reduce inter-relay-interference.
 10. Thesystem of claim 1, wherein the processors cause the pluarlity of relaynodes to apply a training sequence at each relay node, and to multiplexthe training sequences.
 11. A method for disintegrated channelestimation in a wireless communication network, comprising the steps of:receiving a pilot signal and a data signal at a pluarlity of relay nodesin communication with a source node; filtering the pilot signal at eachof the plurality of relay nodes to generate a filtered pilot signal;process the filtered pilot signal at each of the plurality of relaynodes to generate an orthogonal pilot signal; superimposing theorthogonal pilot signal onto the filtered pilot signal at each of thepluarlity of relay nodes to generate a superimposed filtered pilotsignal; and transmitting the superimposed filtered pilot signals fromthe plurality of relay nodes to the destination node, wherein thedestination node receives the superimposed filtered pilot signals fromthe plurality of wireless realy nodes, and estimates disintegratedchannel information using the superimposed filtered pilot signals. 12.The method of claim 11, further comprising processing the data signal ateach of the pluarlity of relay nodes into a space-time coded datasignal.
 13. The method of claim 12, further transmitting the space-timecoded data signals from the pluarlity of relay nodes to the destinationnode.
 14. The method of claim 13, further comprising receiving thespace-time coded data signals at the destination node and processing thespace-time coded data signals using the disintegrated channelinformation.
 15. The method of claim 11, further comprising opeartingthe plurality of relay nodes in a filter-and-forward mode of operation.16. The method of claim 15, further comprising communicating between theplurality of relay nodes and the source node and the destination nodeusing time-division multiplexing (TDM).
 17. The method of claim 15,further comprising communicating between the plurality of relay nodescommunicate and the source node and the destination node usingfrequency-division multiplexing (FDM).
 18. The method of claim 15,further comprising communicating between the plurality of relay nodescommunicate with the source node and the destination node usingcode-division multiplexing (CDM).
 19. The method of claim 15, furthercomprising performing filtering at the pluarlity of relay nodes toreduce inter-relay-interference.
 20. The method of claim 11, furthercomprising applying a training sequence at each relay node andmultiplexing the training sequences.